reserve G for non empty addLoopStr;
reserve x for Element of G;
reserve M for non empty MidStr;
reserve p,q,r for Point of M;
reserve w for Function of [:the carrier of M,the carrier of M:], the carrier
  of G;
reserve S for non empty set;
reserve a,b,b9,c,c9,d for Element of S;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve x for Element of G;
reserve w for Function of [:S,S:],the carrier of G;

theorem Th6:
  w is_atlas_of S,G implies for b,x ex a st w.(a,b) = x
proof
  assume
A1: w is_atlas_of S,G;
  let b,x;
  consider a such that
A2: w.(b,a) = -x by A1;
  take a;
  w.(a,b) = -(-x) by A1,A2,Th4
    .= x by RLVECT_1:17;
  hence thesis;
end;
